/// <summary>
/// This method saves some time if both the inverse and the determinant is needed by only having to compute
/// the inverse once.
/// </summary>
public static void InverseDeterminant(DualMatrix matrix, out DualMatrix inverse, out DualNumber determinant)
{
int m = matrix.Rows;
if (matrix.Columns != m)
{
throw new ArgumentException("The matrix isn't a square matrix.");
}
if (m <= 3)
{
// The naive implementation seems to be faster (for all values of n). Maybe it's
// faster to perform the LU decomposition directly on the DualNumbers?
determinant = DeterminantDirect(m, matrix);
inverse = InverseDirect(m, matrix, determinant);
return;
}
int n = GradientLength(m, matrix);
LUDecomposition lu = LUDecomposition.Decompose(matrix.GetValues());
Matrix inverse0 = lu.Inverse();
double determinant0 = lu.Determinant();
inverse = Inverse(matrix, m, n, inverse0);
determinant = Determinant(matrix, m, n, inverse0, determinant0);
}
public static void InverseDeterminant(DualMatrix matrix, out DualMatrix inverse, out DualNumber determinant)
{
int n = matrix.Rows;
if (matrix.Columns != n)
{
throw new ArgumentException("The matrix is not a square matrix.");
}
DualNumber[,] a = matrix.ToArray();
if (!spdmatrixcholesky(ref a, n, false))
{
throw new ArithmeticException();
}
determinant = spdmatrixcholeskydet(ref a, n);
int info = 0;
spdmatrixcholeskyinverse(ref a, n, false, ref info);
for (int i = 0; i < n; i++)
{
for (int j = i + 1; j < n; j++)
{
a[i, j] = a[j, i];
}
}
inverse = new DualMatrix(a);
}
public static DualMatrix Inverse(DualMatrix matrix)
{
int n = matrix.Rows;
if (matrix.Columns != n)
{
throw new ArgumentException("The matrix is not a square matrix.");
}
DualNumber[,] a = matrix.ToArray();
int info = 0;
spdmatrixinverse(ref a, n, false, ref info);
if (info != 1)
{
throw new ArithmeticException();
}
for (int i = 0; i < n; i++)
{
for (int j = i + 1; j < n; j++)
{
a[i, j] = a[j, i];
}
}
return new DualMatrix(a);
}
public DualVector(DualNumber[] entries)
{
int n = entries.Length;
DualNumber[,] a = new DualNumber[n, 1];
for (int i = 0; i < n; i++)
{
a[i, 0] = entries[i];
}
inner = new DualMatrix(a);
}
public static DualMatrix Inverse(DualMatrix matrix)
{
int m = matrix.Rows;
if (matrix.Columns != m)
{
throw new ArgumentException("The matrix isn't a square matrix.");
}
if (m <= 3)
{
DualNumber determinant = DeterminantDirect(m, matrix);
return InverseDirect(m, matrix, determinant);
}
int n = GradientLength(m, matrix);
return Inverse(matrix, m, n, LUDecomposition.Inverse(matrix.GetValues()));
}
public static DualMatrix Transpose(DualMatrix matrix)
{
if (matrix == null)
{
throw new ArgumentNullException();
}
int n = matrix.Columns;
int m = matrix.Rows;
DualMatrix b = new DualMatrix(n, m);
for (int i = 0; i < n; i++)
{
for (int j = 0; j < m; j++)
{
b[i, j] = matrix[j, i];
}
}
return b;
}
public static DualNumber Trace(DualMatrix matrix)
{
if (matrix == null)
{
throw new ArgumentNullException();
}
int n = matrix.Rows;
if (matrix.Columns != n)
{
throw new ArgumentException("Non-square matrix.");
}
DualNumber s = 0.0;
for (int i = 0; i < n; i++)
{
s += matrix[i, i];
}
return s;
}
public static DualMatrix Inverse(DualMatrix matrix)
{
// Use direct computation for low dimenstions; LU decomposition otherwise.
return DualLUDecomposition.Inverse(matrix);
}
public static DualMatrix Identity(int rows, int columns)
{
if (rows < 0 || columns < 0)
{
throw new ArgumentOutOfRangeException();
}
int n = Math.Min(rows, columns);
DualMatrix a = new DualMatrix(rows, columns);
for (int i = 0; i < n; i++)
{
a[i, i] = 1.0;
}
return a;
}
public static DualMatrix Diagonal(params DualNumber[] values)
{
if (values == null)
{
throw new ArgumentNullException();
}
int n = values.Length;
DualMatrix a = new DualMatrix(n, n);
for (int i = 0; i < n; i++)
{
a[i, i] = values[i];
}
return a;
}
private static DualMatrix InverseDirect(int m, DualMatrix matrix, DualNumber determinant)
{
if (m == 1)
{
DualNumber a11 = matrix[0, 0];
return new DualMatrix(new DualNumber[,] { { 1.0 / a11 } });
}
else if (m == 2)
{
DualNumber a11 = matrix[0, 0];
DualNumber a12 = matrix[0, 1];
DualNumber a21 = matrix[1, 0];
DualNumber a22 = matrix[1, 1];
return new DualMatrix(new DualNumber[,] { { a22 / determinant, -a12 / determinant }, { -a21 / determinant, a11 / determinant } });
}
else if (m == 3)
{
DualNumber a11 = matrix[0, 0];
DualNumber a12 = matrix[0, 1];
DualNumber a13 = matrix[0, 2];
DualNumber a21 = matrix[1, 0];
DualNumber a22 = matrix[1, 1];
DualNumber a23 = matrix[1, 2];
DualNumber a31 = matrix[2, 0];
DualNumber a32 = matrix[2, 1];
DualNumber a33 = matrix[2, 2];
// Using http://mathworld.wolfram.com/MatrixInverse.html.
return new DualMatrix(new DualNumber[,] {
{ (a22 * a33 - a23 * a32) / determinant, (a13 * a32 - a12 * a33) / determinant, (a12 * a23 - a13 * a22) / determinant },
{ (a23 * a31 - a21 * a33) / determinant, (a11 * a33 - a13 * a31) / determinant, (a13 * a21 - a11 * a23) / determinant },
{ (a21 * a32 - a22 * a31) / determinant, (a12 * a31 - a11 * a32) / determinant, (a11 * a22 - a12 * a21) / determinant }
});
}
else
{
throw new NotImplementedException();
}
}
public DualMatrix SetMatrix(int row, int column, DualMatrix subMatrix)
{
if (row < 0 || row + subMatrix.Rows > Rows || column < 0 || column + subMatrix.Columns > Columns)
{
throw new ArgumentOutOfRangeException();
}
int n = subMatrix.Rows;
int m = subMatrix.Columns;
DualMatrix b = new DualMatrix(entries);
for (int i = 0; i < n; i++)
{
for (int j = 0; j < m; j++)
{
b[row + i, column + j] = subMatrix[i, j];
}
}
return b;
}
public DualVector(Vector values, Vector[] gradients, Vector[,] hessians)
{
int n = 0;
Matrix[] matrixGradients = null;
if (gradients != null)
{
n = gradients.Length;
matrixGradients = new Matrix[n];
for (int i = 0; i < n; i++)
{
if (gradients[i] == null)
{
throw new ArgumentNullException("gradients", "The gradients must be fully specified.");
}
matrixGradients[i] = (Matrix)gradients[i];
}
}
Matrix[,] matrixHessians = null;
if (hessians != null)
{
if (hessians.GetLength(0) != n || hessians.GetLength(1) != n)
{
throw new ArgumentException("Inconsistent number of derivatives.");
}
matrixHessians = new Matrix[n, n];
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
if (hessians[i, j] == null)
{
throw new ArgumentNullException("hessians", "The Hessians must be fully specified.");
}
matrixHessians[i, j] = (Matrix)hessians[i, j];
}
}
}
inner = new DualMatrix((Matrix)values, matrixGradients, matrixHessians);
}
public static DualMatrix operator -(DualMatrix a, DualMatrix b)
{
int n = a.Rows;
int m = a.Columns;
if (b.Rows != n || b.Columns != m)
{
throw new ArgumentException("The matrix dimensions don't match.");
}
DualMatrix c = new DualMatrix(n, m);
for (int i = 0; i < n; i++)
{
for (int j = 0; j < m; j++)
{
c[i, j] = a[i, j] - b[i, j];
}
}
return c;
}
public static DualMatrix operator *(DualMatrix a, DualMatrix b)
{
int n = a.Rows;
int m = b.Columns;
int l = a.Columns;
if (b.Rows != l)
{
throw new ArgumentException("The matrix dimensions don't match.");
}
DualMatrix c = new DualMatrix(n, m);
for (int i = 0; i < n; i++)
{
for (int j = 0; j < m; j++)
{
DualNumber s = 0.0;
for (int k = 0; k < l; k++)
{
s += a[i, k] * b[k, j];
}
c[i, j] = s;
}
}
return c;
}
private static DualNumber Determinant(DualMatrix matrix, int m, int n, Matrix inverse, double determinant)
{
if (n < 0)
{
return determinant;
}
// Compute the gradient in the first loop (not scaled by the determinant yet).
// These are used to compute the Hessian.
double[] gradientArray = new double[n];
for (int i = 0; i < n; i++)
{
// Formula (41) in The Matrix Cookbook.
double s = 0.0;
for (int k = 0; k < m; k++)
{
for (int l = 0; l < m; l++)
{
s += inverse[k, l] * matrix[l, k].Gradient[i];
}
}
gradientArray[i] = s;
}
// Compute the Hessian.
double[] hessianArray = new double[n * (n + 1) / 2];
for (int i = 0, h = 0; i < n; i++)
{
for (int j = i; j < n; j++, h++)
{
// Formula (42) in The Matrix Cookbook. Also works when taking partial derivatives
// with respect to two different variables.
double s = 0.0;
for (int k = 0; k < m; k++)
{
for (int l = 0; l < m; l++)
{
double si = 0.0;
double sj = 0.0;
for (int q = 0; q < m; q++)
{
si += inverse[k, q] * matrix[q, l].Gradient[i];
sj += inverse[l, q] * matrix[q, k].Gradient[j];
}
s += inverse[k, l] * matrix[l, k].Hessian[i, j] - si * sj;
}
}
hessianArray[h] = determinant * (gradientArray[i] * gradientArray[j] + s);
}
// Final scaling of the gradient. This index is not used anymore in the loop.
gradientArray[i] *= determinant;
}
// Finally create the DualNumber instance.
return new DualNumber(determinant, gradientArray, hessianArray);
}
private static int GradientLength(int m, DualMatrix matrix)
{
int n = -1;
for (int i = 0; i < m; i++)
{
for (int j = 0; j < m; j++)
{
Vector gradient = matrix[i, j].Gradient;
if (gradient != null)
{
if (n != -1 && gradient.Length != n)
{
throw new ArgumentException("Inconsistent number of derivatives.");
}
n = gradient.Length;
}
}
}
return n;
}
public DualMatrix GetMatrix(int row, int column, int rows, int columns)
{
if (rows < 0 || row < 0 || row + rows > Rows || columns < 0 || column < 0 || column + columns > Columns)
{
throw new ArgumentOutOfRangeException();
}
int n = rows;
int m = columns;
DualMatrix a = new DualMatrix(n, m);
for (int i = 0; i < n; i++)
{
for (int j = 0; j < m; j++)
{
a[i, j] = this[row + i, column + j];
}
}
return a;
}
private static DualNumber DeterminantDirect(int m, DualMatrix matrix)
{
if (m == 1)
{
DualNumber a11 = matrix[0, 0];
return a11;
}
else if (m == 2)
{
DualNumber a11 = matrix[0, 0];
DualNumber a12 = matrix[0, 1];
DualNumber a21 = matrix[1, 0];
DualNumber a22 = matrix[1, 1];
return a11 * a22 - a12 * a21;
}
else if (m == 3)
{
DualNumber a11 = matrix[0, 0];
DualNumber a12 = matrix[0, 1];
DualNumber a13 = matrix[0, 2];
DualNumber a21 = matrix[1, 0];
DualNumber a22 = matrix[1, 1];
DualNumber a23 = matrix[1, 2];
DualNumber a31 = matrix[2, 0];
DualNumber a32 = matrix[2, 1];
DualNumber a33 = matrix[2, 2];
return a11 * a22 * a33 + a12 * a23 * a31 + a13 * a21 * a32 - a11 * a23 * a32 - a12 * a21 * a33 - a13 * a22 * a31;
}
else
{
throw new NotImplementedException();
}
}
public DualMatrix SetEntry(int row, int column, DualNumber value)
{
if (row < 0 || row >= Rows || column < 0 || column >= Columns)
{
throw new ArgumentOutOfRangeException();
}
DualMatrix a = new DualMatrix(entries);
a[row, column] = value;
return a;
}
public static DualNumber Determinant(DualMatrix matrix)
{
// Use direct computation for low dimenstions; LU decomposition otherwise.
return DualLUDecomposition.Determinant(matrix);
}
public static DualMatrix operator -(DualNumber t, DualMatrix a)
{
int n = a.Rows;
int m = a.Columns;
DualMatrix b = new DualMatrix(n, m);
for (int i = 0; i < n; i++)
{
for (int j = 0; j < m; j++)
{
b[i, j] = t - a[i, j];
}
}
return b;
}
private static DualMatrix Inverse(DualMatrix matrix, int m, int n, Matrix inverse)
{
if (n < 0)
{
// Return a matrix without derivatives information (using the implicit matrix conversion).
return inverse;
}
double[,][] gradientArrays = new double[m, m][];
// Compute all gradients in the first loop. These are used to compute the Hessian.
for (int i0 = 0; i0 < m; i0++)
{
for (int j0 = 0; j0 < m; j0++)
{
double[] gradientArray = new double[n];
for (int i = 0; i < n; i++)
{
// Formula (36) in The Matrix Cookbook.
double s = 0.0;
for (int k = 0; k < m; k++)
{
for (int l = 0; l < m; l++)
{
s += inverse[i0, k] * matrix[k, l].Gradient[i] * inverse[l, j0];
}
}
gradientArray[i] = -s;
}
gradientArrays[i0, j0] = gradientArray;
}
}
// Compute Hessians and DualNumber instances.
DualNumber[,] a = new DualNumber[m, m];
for (int i0 = 0; i0 < m; i0++)
{
for (int j0 = 0; j0 < m; j0++)
{
double[] hessianArray = new double[n * (n + 1) / 2];
for (int i = 0, h = 0; i < n; i++)
{
for (int j = i; j < n; j++, h++)
{
double s = 0.0;
for (int k = 0; k < m; k++)
{
for (int l = 0; l < m; l++)
{
s += gradientArrays[i0, k][i] * matrix[k, l].Gradient[j] * inverse[l, j0]
+ inverse[i0, k] * matrix[k, l].Gradient[h] * inverse[l, j0]
+ inverse[i0, k] * matrix[k, l].Gradient[j] * gradientArrays[l, j0][i];
}
}
hessianArray[h] = -s;
}
}
a[i0, j0] = new DualNumber(inverse[i0, j0], gradientArrays[i0, j0], hessianArray);
}
}
return new DualMatrix(a);
}
private DualVector(DualMatrix inner)
{
if (inner.Columns != 1)
{
throw new ArgumentException();
}
this.inner = inner;
}
public static DualMatrix Basis(int rows, int columns, int row, int column)
{
if (rows < 0 || row < 0 || row >= rows || columns < 0 || column < 0 || column >= columns)
{
throw new ArgumentOutOfRangeException();
}
DualMatrix a = new DualMatrix(rows, columns);
a[row, column] = 1.0;
return a;
}